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August  2018, 15(4): 883-904. doi: 10.3934/mbe.2018040

Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model

Moitri Sen 1, , Malay Banerjee 2,, and  Yasuhiro Takeuchi 3,

1. 

Department of Mathematics, National Institute of Technology, Patna, Bihar-800005, India
2. 

Department of Mathematics & Statistics, Indian Institute of Technology Kanpur, Uttar Pradesh-208016, India
3. 

Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa, Japan

* Corresponding authorr: malayb@iitk.ac.in.

Received  June 26, 2017 Accepted  November 25, 2017 Published  March 2018

One of the important ecological challenges is to capture the complex dynamics and understand the underlying regulating ecological factors. Allee effect is one of the important factors in ecology and taking it into account can cause significant changes to the system dynamics. In this work we consider a two prey-one predator model where the growth of both the prey population is subjected to Allee effect, and the predator is generalist as it survives on both the prey populations. We analyze the role of Allee effect on the dynamics of the system, knowing the dynamics of the model without Allee effect. Interestingly we have observed through a comprehensive bifurcation study that incorporation of Allee effect enriches the local as well as the global dynamics of the system. Specially after a certain threshold value of the Allee effect, it has a very significant effect on the chaotic dynamics of the system. In course of the bifurcation analysis we have explored all possible bifurcations such as the existence of transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation and period-doubling route to chaos respectively.

Keywords: Prey-predator, Allee effect, stability, bifurcation, chaos.
Mathematics Subject Classification: Primary: 34D20, 34F10; Secondary: 92D25.
Citation: Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040
References:
[1]

P. Aguirre, E. González-Olivares and E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69 (2009), 1244-1262.  doi: 10.1137/070705210.  Google Scholar

[2]

P. Aguirre, E. González-Olivares and E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10 (2009), 1401-1416.  doi: 10.1016/j.nonrwa.2008.01.022.  Google Scholar

[3]

W. C. Allee, Animal Aggregations: A study in general sociology, The Quarterly Review of Biology, 2 (1927), 367-398.  doi: 10.1086/394281.  Google Scholar

[4]

L. Berec, E. Angulo and F. Courchamp, Multiple Allee effects and population management, Trends in Ecology & Evolution, 22 (2007), 185-191.  doi: 10.1016/j.tree.2006.12.002.  Google Scholar

[5]

F. Berezovskaya, S. Wirkus, B. Song and C. Castillo-Chavez, Dynamics of population communities with prey migrations and Allee effects: a bifurcation approach, Mathematical Medicine and Biology, 28 (2011), 129-152.  doi: 10.1093/imammb/dqq022.  Google Scholar

[6]

E. D. Conway and J. A. Smoller, Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46 (1986), 630-642.  doi: 10.1137/0146043.  Google Scholar

[7]

F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology & Evolution, 14 (1999), 405-410.  doi: 10.1016/S0169-5347(99)01683-3.  Google Scholar

[8]

B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.  doi: 10.1111/j.1939-7445.1989.tb00119.x.  Google Scholar

[9]

Y. C. Lai and R. L. Winslow, Geometric properties of the chaotic saddle responsible for supertransients in spatiotemporal chaotic systems Physical Review Letters, 74 (1995), p5208. doi: 10.1103/PhysRevLett.74.5208.  Google Scholar

[10]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158.  doi: 10.1006/tpbi.1993.1007.  Google Scholar

[11]

A. J. Lotka, A Natural Population Norm I & Ⅱ, 1913. Google Scholar

[12]

A. Morozov, S. Petrovskii and B.-L. Li, Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect, Journal of Theoretical Biology, 238 (2006), 18-35.  doi: 10.1016/j.jtbi.2005.05.021.  Google Scholar

[13]

A. Y. Morozov, M. Banerjee and S. V. Petrovskii, Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect, Journal of Theoretical Biology, 396 (2016), 116-124.  doi: 10.1016/j.jtbi.2016.02.016.  Google Scholar

[14]

M. Sen, M. Banerjee and A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecological Complexity, 11 (2012), 12-27.  doi: 10.1016/j.ecocom.2012.01.002.  Google Scholar

[15]

M. Sen and M. Banerjee, Rich global dynamics in a prey-predator model with Allee effect and density dependent death rate of predator, International Journal of Bifurcation and Chaos(1530007), 25 (2015), 17pp.   Google Scholar

[16]

P. A. Stephens and W. J. Sutherland, Consequences of the allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.  Google Scholar

[17]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, 1996.  Google Scholar

[18]

Y. Takeuchi and N. Adachi, Existence and bifurcation of stable equilibrium in two-prey, one-predator communities, Bulletin of Mathematical Biology, 45 (1983), 877-900.  doi: 10.1007/BF02458820.  Google Scholar

[19]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560.  doi: 10.1038/118558a0.  Google Scholar

[20]

G. Wang, X. G. Liang and F. Z. Wang, The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124 (1999), 183-192.  doi: 10.1016/S0304-3800(99)00160-X.  Google Scholar

[21]

J. Zu and M. Mimura, The impact of Allee effect on a predator-prey system with Holling type ii functional response, Applied Mathematics and Computation, 217 (2010), 3542-3556.  doi: 10.1016/j.amc.2010.09.029.  Google Scholar

show all references

References:
[1]

P. Aguirre, E. González-Olivares and E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69 (2009), 1244-1262.  doi: 10.1137/070705210.  Google Scholar

[2]

P. Aguirre, E. González-Olivares and E. Sáez, Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10 (2009), 1401-1416.  doi: 10.1016/j.nonrwa.2008.01.022.  Google Scholar

[3]

W. C. Allee, Animal Aggregations: A study in general sociology, The Quarterly Review of Biology, 2 (1927), 367-398.  doi: 10.1086/394281.  Google Scholar

[4]

L. Berec, E. Angulo and F. Courchamp, Multiple Allee effects and population management, Trends in Ecology & Evolution, 22 (2007), 185-191.  doi: 10.1016/j.tree.2006.12.002.  Google Scholar

[5]

F. Berezovskaya, S. Wirkus, B. Song and C. Castillo-Chavez, Dynamics of population communities with prey migrations and Allee effects: a bifurcation approach, Mathematical Medicine and Biology, 28 (2011), 129-152.  doi: 10.1093/imammb/dqq022.  Google Scholar

[6]

E. D. Conway and J. A. Smoller, Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46 (1986), 630-642.  doi: 10.1137/0146043.  Google Scholar

[7]

F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology & Evolution, 14 (1999), 405-410.  doi: 10.1016/S0169-5347(99)01683-3.  Google Scholar

[8]

B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.  doi: 10.1111/j.1939-7445.1989.tb00119.x.  Google Scholar

[9]

Y. C. Lai and R. L. Winslow, Geometric properties of the chaotic saddle responsible for supertransients in spatiotemporal chaotic systems Physical Review Letters, 74 (1995), p5208. doi: 10.1103/PhysRevLett.74.5208.  Google Scholar

[10]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158.  doi: 10.1006/tpbi.1993.1007.  Google Scholar

[11]

A. J. Lotka, A Natural Population Norm I & Ⅱ, 1913. Google Scholar

[12]

A. Morozov, S. Petrovskii and B.-L. Li, Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect, Journal of Theoretical Biology, 238 (2006), 18-35.  doi: 10.1016/j.jtbi.2005.05.021.  Google Scholar

[13]

A. Y. Morozov, M. Banerjee and S. V. Petrovskii, Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect, Journal of Theoretical Biology, 396 (2016), 116-124.  doi: 10.1016/j.jtbi.2016.02.016.  Google Scholar

[14]

M. Sen, M. Banerjee and A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecological Complexity, 11 (2012), 12-27.  doi: 10.1016/j.ecocom.2012.01.002.  Google Scholar

[15]

M. Sen and M. Banerjee, Rich global dynamics in a prey-predator model with Allee effect and density dependent death rate of predator, International Journal of Bifurcation and Chaos(1530007), 25 (2015), 17pp.   Google Scholar

[16]

P. A. Stephens and W. J. Sutherland, Consequences of the allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.  Google Scholar

[17]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, 1996.  Google Scholar

[18]

Y. Takeuchi and N. Adachi, Existence and bifurcation of stable equilibrium in two-prey, one-predator communities, Bulletin of Mathematical Biology, 45 (1983), 877-900.  doi: 10.1007/BF02458820.  Google Scholar

[19]

V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560.  doi: 10.1038/118558a0.  Google Scholar

[20]

G. Wang, X. G. Liang and F. Z. Wang, The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124 (1999), 183-192.  doi: 10.1016/S0304-3800(99)00160-X.  Google Scholar

[21]

J. Zu and M. Mimura, The impact of Allee effect on a predator-prey system with Holling type ii functional response, Applied Mathematics and Computation, 217 (2010), 3542-3556.  doi: 10.1016/j.amc.2010.09.029.  Google Scholar

Figure 1.  Positions of the nullclines projected on the xy-plane showing the feasibility of $E_5$.
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Figure 2.  Schematic bifurcation diagram for the model (5) in $\alpha_1\, \alpha_2$-parametric space. Transcritical bifurcation curves (violet and magenta), saddle-node bifurcation curve(s) (black, blue and cyan), Hopf-bifurcation curve (yellow and green) and the red curve for the first period doubling bifurcation for limit cycle divide the parametric space into seventeen regions ($R_1\, \rightarrow\, R_{17}$). Point marked in black colour is Bogdanov-Takens bifurcation point, point of tangency of transcritical bifurcation curve for $E_*$ and the saddle node bifurcation curve for $E_5$ is marked with a blue dot, and the point of tangency transcritical bifurcation curve and the saddle node bifurcation curve for $E_*$ is marked with a red dot. Stability properties of various equilibria with different parametric regions are summarized at Table-1.
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Figure 5.  Schematic bifurcation diagram for the model (5) in $\alpha_1\, \alpha_2$-parametric space. Transcritical bifurcation curves (violet and magenta), saddle-node bifurcation curve(s) (black, blue and cyan), Hopf-bifurcation curve (yellow and green) and the red curve for the first period doubling bifurcation for limit cycle divide the parametric space into sixteen regions ($R_1\, \rightarrow\, R_{16}$) and three more regions $R_{4A}, \, R_{5A}\, R_{6A}$. Point marked in black colour is Bogdanov-Takens bifurcation point, point of tangency of transcritical bifurcation curve for $E_*$ and the saddle node bifurcation curve for $E_5$ are marked with a blue dot and the point of tangency transcritical bifurcation curve and the saddle node bifurcation curve for $E_*$ is marked with a red dot. Stability properties of various equilibria with different parametric regions are summarized at Table-1.
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Figure 3.  Bifurcation diagram with respect to the parameter $\alpha_1$, other parameter values are $\alpha = 1, \alpha_2 = 0.01, \beta = 1.5, \beta_1 = 2, \beta_2 = 2, \beta_3 = 1, \gamma = 1, d = 0.5, \mu = 1, \epsilon = 5$. $\alpha_1\in[0, 0.0082], \, [0.0083, 0.0118]$ and $[0.0119, 0.0125]$ correspond to regions $R_4\, R_5$ and $R_6$ respectively. $x$-components of $E_0, E_3, E_5, E_{1*}, E_{2*}$ are marked in blue, green, red, magenta, black colours in Fig 2 respectively. Continuous line represents stability of concerned equilibrium point when $\alpha_1$ increases. $E_{2*}$ loses stability through Hopf-bifurcation at $\alpha_1\equiv \alpha_{1H} = 0.0083$, first period doubling occurs at $\alpha_1 = 0.01185$, chaotic dynamics is observed for $\alpha_1\in[0.0125, 0.0135]$.
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Figure 4.  Peak-adding bifurcation: successive peaks appear as the supplementary local maxima and minima occur in (c), (d) and (e) for $\alpha_1=0.0121,0.0122$ and 0.0123 respectively.
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Figure 6.  Bifurcation diagram with respect to the parameter $\alpha_2$, other parameter values are $\alpha = 1, \alpha_1 = 0.005, \beta = 1.5, \beta_1 = 2, \beta_2 = 2, \beta_3 = 1, \gamma = 1, d = 0.5, \mu = 1, \epsilon = 10$. $\alpha_2\in[0.05423, 0.056], \, [0.056, 0.0642]$ and $[0.0643, 0.07]$ correspond to regions $R_4\, R_5$ and $R_6$ respectively. $x$-components of $E_0, E_3, E_5, E_{1*}, E_{2*}$ are marked in blue, green, red, magenta, black colours respectively Fig 5. Continuous line represents stability of concerned equilibrium point when $\alpha_2$ decreases. $E_{2*}$ loses stability through Hopf-bifurcation at $\alpha_2\equiv \alpha_{2h} = 0.0642$, first period doubling occurs at $\alpha_2 = 0.0577$, chaotic dynamics is observed for $\alpha_2\in[0.05423, 0.056]$.
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Table 1.  Summary of existence and stability conditions for the equilibria of (5).
Equilibrium Existence Stability
$\displaystyle E_0(0, 0, 0)$ Always LAS
$\displaystyle E^+_1(+, 0, 0)$ $\beta_1\geq(1+\sqrt{\alpha_1})^2$ LAS if $x^+_1<\frac{\beta_3}{d \epsilon}$, Saddle point if $x^+_1>\frac{\beta_3}{d \epsilon}$
$\displaystyle E^-_1(+, 0, 0)$ $\beta_1\geq(1+\sqrt{\alpha_1})^2$ Saddle point with one dimensional unstable manifold if $x^-_1<\frac{\beta_3}{d \epsilon}$, Saddle point with two dimensional unstable manifolds $x^-_1>\frac{\beta_3}{d \epsilon}$
$\displaystyle E^+_2(0, +, 0)$ $\beta_2\geq(\sqrt{\gamma}+\sqrt{\alpha_2})^2$ LAS if $y^+_2<\frac{\beta_3}{d \mu}$, Saddle point if $y^+_2>\frac{\beta_3}{d \mu}$
$\displaystyle E^-_2(0, +, 0)$ $\beta_2\geq(\sqrt{\gamma}+\sqrt{\alpha_2})^2$ Saddle point with one dimensional unstable manifold if $y^-_2<\frac{\beta_3}{d \mu}$, Saddle point with two dimensional unstable manifolds if $y^-_2>\frac{\beta_3}{d \mu}$.
$\displaystyle E_3(+, 0, +)$ $d\beta_1\beta_3\epsilon>(\beta_3+d\epsilon)(\beta_3+d\alpha_1\epsilon)$ LAS if $(x_3+\alpha_1)^2>\beta_1\alpha_1$ otherwise a saddle point
$\displaystyle E_4(0, +, +)$ $d\beta_2\beta_3\mu>(\beta_3+d\mu\gamma)(\beta_3+d\alpha_2\mu)$ LAS if $(y_4+\alpha_2)^2>\beta_2\alpha_2$ otherwise a saddle point
$\displaystyle E_5(+, +, 0)$ See proposition 6 See proposition 6
$\displaystyle E_*(+, +, +)$ See proposition 7 See proposition 7
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Equilibrium Existence Stability
$\displaystyle E_0(0, 0, 0)$ Always LAS
$\displaystyle E^+_1(+, 0, 0)$ $\beta_1\geq(1+\sqrt{\alpha_1})^2$ LAS if $x^+_1<\frac{\beta_3}{d \epsilon}$, Saddle point if $x^+_1>\frac{\beta_3}{d \epsilon}$
$\displaystyle E^-_1(+, 0, 0)$ $\beta_1\geq(1+\sqrt{\alpha_1})^2$ Saddle point with one dimensional unstable manifold if $x^-_1<\frac{\beta_3}{d \epsilon}$, Saddle point with two dimensional unstable manifolds $x^-_1>\frac{\beta_3}{d \epsilon}$
$\displaystyle E^+_2(0, +, 0)$ $\beta_2\geq(\sqrt{\gamma}+\sqrt{\alpha_2})^2$ LAS if $y^+_2<\frac{\beta_3}{d \mu}$, Saddle point if $y^+_2>\frac{\beta_3}{d \mu}$
$\displaystyle E^-_2(0, +, 0)$ $\beta_2\geq(\sqrt{\gamma}+\sqrt{\alpha_2})^2$ Saddle point with one dimensional unstable manifold if $y^-_2<\frac{\beta_3}{d \mu}$, Saddle point with two dimensional unstable manifolds if $y^-_2>\frac{\beta_3}{d \mu}$.
$\displaystyle E_3(+, 0, +)$ $d\beta_1\beta_3\epsilon>(\beta_3+d\epsilon)(\beta_3+d\alpha_1\epsilon)$ LAS if $(x_3+\alpha_1)^2>\beta_1\alpha_1$ otherwise a saddle point
$\displaystyle E_4(0, +, +)$ $d\beta_2\beta_3\mu>(\beta_3+d\mu\gamma)(\beta_3+d\alpha_2\mu)$ LAS if $(y_4+\alpha_2)^2>\beta_2\alpha_2$ otherwise a saddle point
$\displaystyle E_5(+, +, 0)$ See proposition 6 See proposition 6
$\displaystyle E_*(+, +, +)$ See proposition 7 See proposition 7
Table 2.  Here $E_3$ undergoes a subcritical Hopf-bifurcation and $E_{2*}$ looses stability through supercritical Hopf-bifurcation. The Hopf bifurcating limit cycle around $E_{2*}$ disappears through chaos.
Region Feasible Equilibria Attractors
$\displaystyle R_1$ $E_0, E_1^+, E_1^-, E_3$ $E_0, E_3$
$\displaystyle R_2$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3$ $E_0, E_2^+, E_3$
$\displaystyle R_3$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2$ $E_0, E_2^+, E_3$
$\displaystyle R_4$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{1*}, E_{2*}$ $E_0, E_2^+, E_3, E_{2*}$
$\displaystyle R_5$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{1*}, E_{2*}$, $E_0, E_2^+, E_3$ & stable limit $E_{1*}, E_{2*}$ cycle around $E_{2*}$
$\displaystyle R_6$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{1*}, E_{2*}$ $E_0, E_2^+, E_3$
$\displaystyle R_7$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{2*}$ $E_0, E_2^+, E_3$
$\displaystyle R_8$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{2*}$ $E_0, E_2^+$
$\displaystyle R_9$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2$ $E_0, E_2^+$
$\displaystyle R_{10}$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_5^1, E_5^2$ $E_0, E_2^+$
$\displaystyle R_{11}$ $E_0, E_2^+, E_2^-, E_5^1, E_5^2$ $E_0, E_2^+$
$\displaystyle R_{12}$ $E_0, E_2^+, E_2^-$ $E_0, E_2^+$
$\displaystyle R_{13}$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-$ $E_0, E_2^+$
$\displaystyle R_{14}$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3$ $E_0, E_2^+$
$\displaystyle R_{15}$ $E_0, E_1^+, E_1^-, E_3$ $E_0$
$\displaystyle R_{16}$ $E_0, E_1^+, E_1^-$ $E_0$
$\displaystyle R_{17}$ $E_0$ $E_0$
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Region Feasible Equilibria Attractors
$\displaystyle R_1$ $E_0, E_1^+, E_1^-, E_3$ $E_0, E_3$
$\displaystyle R_2$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3$ $E_0, E_2^+, E_3$
$\displaystyle R_3$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2$ $E_0, E_2^+, E_3$
$\displaystyle R_4$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{1*}, E_{2*}$ $E_0, E_2^+, E_3, E_{2*}$
$\displaystyle R_5$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{1*}, E_{2*}$, $E_0, E_2^+, E_3$ & stable limit $E_{1*}, E_{2*}$ cycle around $E_{2*}$
$\displaystyle R_6$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{1*}, E_{2*}$ $E_0, E_2^+, E_3$
$\displaystyle R_7$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{2*}$ $E_0, E_2^+, E_3$
$\displaystyle R_8$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2, E_{2*}$ $E_0, E_2^+$
$\displaystyle R_9$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3, E_5^1, E_5^2$ $E_0, E_2^+$
$\displaystyle R_{10}$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_5^1, E_5^2$ $E_0, E_2^+$
$\displaystyle R_{11}$ $E_0, E_2^+, E_2^-, E_5^1, E_5^2$ $E_0, E_2^+$
$\displaystyle R_{12}$ $E_0, E_2^+, E_2^-$ $E_0, E_2^+$
$\displaystyle R_{13}$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-$ $E_0, E_2^+$
$\displaystyle R_{14}$ $E_0, E_1^+, E_1^-, E_2^+, E_2^-, E_3$ $E_0, E_2^+$
$\displaystyle R_{15}$ $E_0, E_1^+, E_1^-, E_3$ $E_0$
$\displaystyle R_{16}$ $E_0, E_1^+, E_1^-$ $E_0$
$\displaystyle R_{17}$ $E_0$ $E_0$
Table 3.  Here $E_3$ undergoes a subcritical Hopf-bifurcation and $E_{2*}$ looses stability through supercritical Hopf-bifurcation. The Hopf bifurcating limit cycle around $E_{2*}$ disappears through chaos.
Region Feasible Equilibria Attractors
$\displaystyle R_1$ $E_0,E_1^+,E_1^-,E_3$ $E_0,E_3$
$\displaystyle R_2$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3$ $E_0,E_2^+,E_3$
$\displaystyle R_3$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$ $E_0,E_2^+,E_3$
$\displaystyle R_4$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$ $E_0,E_2^+,E_3,E_{2*}$
$\displaystyle R_5$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$,$E_{1*},E_{2*}$ $E_0,E_2^+,E_3$ stable limit cycle around $E_{2*}$
$\displaystyle R_6$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$,$E_{1*},E_{2*}$ $E_0,E_2^+,E_3$
$\displaystyle R_7$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{2*}$ $E_0,E_2^+,E_3$
$\displaystyle R_8$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{2*}$ $E_0,E_2^+$
$\displaystyle R_9$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$ $E_0,E_2^+$
$\displaystyle R_{10}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_5^1,E_5^2$ $E_0,E_2^+$
$\displaystyle R_{11}$ $E_0,E_2^+,E_2^-$ $E_0,E_2^+$
$\displaystyle R_{12}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-$ $E_0,E_2^+$
$\displaystyle R_{13}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3$ $E_0,E_2^+$
$\displaystyle R_{14}$ $E_0,E_1^+,E_1^-,E_3$ $E_0$
$\displaystyle R_{15}$ $E_0,E_1^+,E_1^-$ $E_0$
$\displaystyle R_{16}$ $E_0$ $E_0$
$\displaystyle R_{6A}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$ $E_0,E_2^+$
$\displaystyle R_{5A}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$, $E_0,E_2^+$ stable limit $E_{1*},E_{2*}$ cycle around $E_{2*}$
$\displaystyle R_{4A}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$ $E_0,E_2^+,E_{2*}$
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Region Feasible Equilibria Attractors
$\displaystyle R_1$ $E_0,E_1^+,E_1^-,E_3$ $E_0,E_3$
$\displaystyle R_2$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3$ $E_0,E_2^+,E_3$
$\displaystyle R_3$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$ $E_0,E_2^+,E_3$
$\displaystyle R_4$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$ $E_0,E_2^+,E_3,E_{2*}$
$\displaystyle R_5$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$,$E_{1*},E_{2*}$ $E_0,E_2^+,E_3$ stable limit cycle around $E_{2*}$
$\displaystyle R_6$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$,$E_{1*},E_{2*}$ $E_0,E_2^+,E_3$
$\displaystyle R_7$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{2*}$ $E_0,E_2^+,E_3$
$\displaystyle R_8$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{2*}$ $E_0,E_2^+$
$\displaystyle R_9$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2$ $E_0,E_2^+$
$\displaystyle R_{10}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_5^1,E_5^2$ $E_0,E_2^+$
$\displaystyle R_{11}$ $E_0,E_2^+,E_2^-$ $E_0,E_2^+$
$\displaystyle R_{12}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-$ $E_0,E_2^+$
$\displaystyle R_{13}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3$ $E_0,E_2^+$
$\displaystyle R_{14}$ $E_0,E_1^+,E_1^-,E_3$ $E_0$
$\displaystyle R_{15}$ $E_0,E_1^+,E_1^-$ $E_0$
$\displaystyle R_{16}$ $E_0$ $E_0$
$\displaystyle R_{6A}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$ $E_0,E_2^+$
$\displaystyle R_{5A}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$, $E_0,E_2^+$ stable limit $E_{1*},E_{2*}$ cycle around $E_{2*}$
$\displaystyle R_{4A}$ $E_0,E_1^+,E_1^-,E_2^+,E_2^-,E_3,E_5^1,E_5^2,E_{1*},E_{2*}$ $E_0,E_2^+,E_{2*}$
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